Stability for a 3D Ladyzhenskaya fluid model with unbounded variable delay

1.   Introduction

The 3D incompressible Navier-Stokes model is expressed as

which was proposed by Navier and Stokes, respectively, for the motion of incompressible viscous fluids with a very small velocity gradient. Since the last century, there have been many interesting results on the existence, uniqueness, regularity and long-time behaviour of the Navier-Stokes equations ^[1,2,3,4,5]. However, for the 3D Navier-Stokes model, the uniqueness of weak solutions and the existence of strong solutions have not been solved.

Mathematicians and physicists have proposed many modified Navier-Stokes models to overcome the difficulty brought about by the nonlinear convection $(u\cdot\nabla)u$, such as the Navier-Stokes-$\alpha$ model, Navier-$\alpha$ model, Navier-Stokes-Voigt model and the globally modified Navier-Stokes model ^[6]. In particular, Ladyzhenskaya ^[7] and Ladyzhenskaya et al. ^[8] also gave a modified Navier-Stokes model. For the Navier-Stokes equations, the velocity derivative should not be very large. Therefore, Ladyzhenskaya et al. ^[8] replaced the Navier-Stokes equations with the following equations:

The model (1.2) can be used to describe the flow of fluid around objects placed in the fluid. Guo and Zhu ^[9] studied the partial regularity of the generalized solutions to model (1.2), and proved that the singular points are concentrated on a closed set whose $5-2q$ dimensional Hausdorff measure is zero when $2 < q \leq \frac 52$. Furthermore, the solution is a regular one when $q > \frac 52$. Recently, da Veiga and Yang ^[10] obtained that the singular points are concentrated on a closed set whose one-dimensional Hausdorff measure is zero when $q < 2$. Regarding the model (1.2), da Veiga has done a series of meaningful works ^[10,11,12].

This model stipulates that for $q > 2$, the stress tensor

is dependent on the symmetrical part $Du$ of the velocity gradient in a polynomial growth manner. As $q = n = 3$ ($n$ is the spatial dimension), (1.2) reduces to the classical Smagorinsky model, which is a turbulence model introduced by Smagorinsky ^[13]. One of the characteristics of the above $q$ growth rate is that one can use the increased coercivity to obtain the existence and uniqueness of solutions. Assuming that $q$ is large enough, the nonlinearity caused by $(u\cdot\nabla) u$ is no longer critical. This overcomes the problem of the Navier-Stokes equation lacking corresponding solvability. In addition, the peculiar features of certain fluids can be better described by using the polynomial growth of the stress tensors (refer to the monograph ^[14]). In order to study some specific types of fluids, and especially to describe shear thickening $(q > 2)$ and shear thinning $(q < 2)$ phenomena, Belloutet et al. ^[15] and Málek et al. ^[16,17] have conducted in-depth research on such models.

However, the uniqueness and stability of solutions for model (1.2) are still open when the Reynolds number is large. With the establishment of the monotonicity method, Lions replaced $Du$ with $\nabla u$ and reduced the model (1.2) to

and

For the system (1.3) or (1.4), Lions ^[18] proved the existence of weak solutions when $q \geq 1 + 2n/(n+2)$. Lions derived the uniqueness of solutions to the system (1.3) when $q \geq (n+2)/2$. However, the uniqueness of the system (1.4) remains an open problem. In particular, Lions also considered a variant of the model (1.3), namely

In this paper, we focus on the model (1.5). Note that it reduces to the classical Navier-Stokes equations comprising (1.1) when $\nu_1 = 0$. Lions ^[18] proved the existence and uniqueness of global weak solutions for the initial-boundary value problem of the model (1.5). Under some assumptions on the external force, the existence of uniform attractors of the model (1.5) was proved in ^[19]. Recently, Yang et al. ^[20] considered the pullback dynamics of (1.5), and presented the finite fractal dimension of pullback attractors. Moreover, the upper semi-continuity of pullback attractors was also studied in ^[20].

The Navier-Stokes equation with hereditary terms was first considered by Caraballo and Real ^[21], and there are many significant results on this model ^{[22,23,24,25]}. Because the current state may be affected by the distant historical state, the delay time is quite large at this time. For the case of infinite delays, it can be seen from the literature ^{[26,27,28,29,30,31]} that the initial value of the delay is usually considered in the following space

Here $H$ is the square-integrable function space satisfying the incompressible condition. At this time, the delayed external force is a distributed delay. That is, it only includes the case of infinite distributed delays. Marín-Rubio et al. ^[27] considered the globally modified Navier-Stokes model with infinite delays, and established the global well-posedness of solutions when the initial value of the delay was in the space $C_\gamma(H)$. Moreover, they proved the exponential stability of stationary solutions. For the 3D Navier-Stokes-Voigt equations in the space $C_\gamma(H)$, Anh and Thanh ^[30] obtained the exponential stability of solutions and proved the existence of global attractors.

The positive constant $\gamma$ plays an important role in the above works. If $\gamma$ disappears, can we get similar results? Liu et al. ^[32] considered the 2D Navier-Stokes models with unbounded variable delays, which is in the following phase space

They established the existence and uniqueness of solutions and analyzed the stability of stationary solutions by using several different methods. $BCL_ {-\infty} (H)$ seems to be a natural relaxation of $C_\gamma (H)$. Recently, Toi ^[33] studied the 3D Navier-Stokes-Voigt equations in $BCL_ {-\infty} (H)$ and gave a sufficient condition for the polynomial stability of stationary solutions.

Inspired by the results ^[27,32,33], this paper is concerned with the stability of solutions to a Ladyzhenskaya model with unbounded variable delays defined on $(0, \infty)\times\Omega$, as follows:

where $\Omega \in \mathbb{R}^3$ is a bounded open domain with the sufficiently smooth boundary $\partial\Omega$, $u = (u_1, u_2, u_3)$ is the velocity field, $p$ is pressure, $\nu_0 > 0$ and $\nu_1 > 0$ are the kinematic viscosities, $f(t)$ is a non-delayed external force and $g(t, u_t)$ is an external force with some hereditary characteristics. The function $u_t$ in the delay term $g(t, u_t)$ is defined on $(-\infty, 0]$ by $u_t(s) = u(t+s), \ s \in (-\infty, 0]$. $\phi$ is the prescribed initial condition. More conditions on $f$ and $\phi$ will be specified later.

The main results of the present paper are summarized as follows:

(I) First, the existence and uniqueness of global weak solutions to the model (1.6) are established by combining Galerkin approximations and the energy method; then, the regularity of weak solutions is obtained. See Theorem 3.1.

(II) Furthermore, we prove the existence and uniqueness of stationary solutions to the model (1.6) by employing a corollary of Schauder's fixed point theorem. For the stability of stationary solutions, the local stability is obtained. Then, by using the Lyapunov function method, we derive that the stationary solution is exponentially stable when the delay length is bounded. See Theorems 4.1, 4.3, and 4.4.

(III) Finally, by virtue of a lemma on the proportional equation, we give a sufficient condition of parameters for the polynomial stability of the stationary solution in a special case of unbounded variable delay. See Theorem 4.6.

2.   Preliminaries

2.1.   Functional spaces, operators and lemmas

Denote

Let $H$ and $V$ be the closures of $E$ in $(L^2(\Omega))^3$ and $(H^1_0(\Omega))^3$, respectively. The inner products in $H$ and $V$ are represented by $(\cdot, \cdot)$ and $((\cdot, \cdot))$, respectively, which are defined as follows:

The associated norms in $H$ and $V$ are represented by $|\cdot|_2$ and $\|\cdot\|$, respectively, which are defined as follows:

Then, we have that $\|u\| = |\nabla u|_2$ for all $u \in V$. It is easy to verify that $H$ and $V$ are Hilbert spaces. Let $H'$ and $V'$ be dual spaces of $H$ and $V$, respectively. We have that $V\hookrightarrow \hookrightarrow H \equiv H' \hookrightarrow V'$, where the injections are dense and continuous. We use $\|\cdot\|_*$ for the norm in $V'$ and $\langle\cdot, \cdot\rangle$ for the dual pairing between $V$ and $V'$, where $\|\cdot\|_*$ is defined as follows:

$P$ denotes the Helmholz-Leray orthogonal projection from $(L^2(\Omega))^3$ onto the space $H$ (see ^[34,35]). We define $A: = -P\Delta$ as the Stokes operator on $D(A) = (H^2(\Omega))^3 \cap V$; then, $A:V\rightarrow V'$ satisfies that $\langle Au, v\rangle = ((u, v))$, and $A$ is an isomorphism from $V$ into $V'$. It holds that $\|Au\|_* = \sup\limits_{v \in V, \|v\| = 1}|\langle Au, v\rangle| = \sup\limits_{v \in V, \|v\| = 1}|((u, v))|\leq\|u\|$, i.e., $\|A\|\leq 1$. Let $\{\lambda_i\}^\infty_{i = 1}$ be the eigenvalues of the operator $A$ with the Dirichlet boundary condition, which satisfy

From the property of the Stokes operator, the corresponding eigenfunctions given by $\{\omega_i\}_{i = 1}^\infty$ form an orthonormal complete basis in $H$. Moreover, we have the following Poincaré inequality

In order to deal with the nonlinear term $\nu_1{\| u\|}^2$, we define the operator $A_1: V\rightarrow V'$ as $A_1u: = -\nu_1\| u\|^2\Delta u$, which satisfies

Obviously, $\langle A_1u-A_1v, u-v\rangle\geq 0$ for any $u, v \in V$. That is, $A_1$ is a monotone operator. We can obtain from (2.1) and (2.3) that

We also introduce the bilinear operator

and the trilinear operator

Furthermore, the bilinear operator ${B(u, v)}$ and the trilinear operator $b(u, v, \omega)$ satisfy the following conditions (see ^[5,34,35]):

where $c_1$ is a constant greater than zero. The following lemma is a refined version of Dini's theorem that has been developed to deal with the delay effect.

Lemma 2.1. (^[29]) (Dini theorem) Suppose that a function sequence $\{S_n (x)\}^\infty_{n = 1}$ on the finite interval $[a, b]$ satisfies

If $\{S_n(x)\} ^\infty_{n = 1}$ is monotonic on $[a, b]$ for any $n \in \mathbb{N}^+$, and $S(x)$ is continuous, then $\{S_n (x)\}^\infty_{n = 1}$ uniformly converges to $S(x)$ on $[a, b]$.

For the fixed-field case, the monotone operator has properties similar to the evolution case (see ^[18]). There is a new type of operator as follows.

Definition 2.2. (^[18]) The operator $K:V\rightarrow V'$ is called the operator of $(M)$ type if $K$ satisfies: supposing the sequence $\{u_m\}\subset V$ such that

and

then, $\chi = K(u)$.

The monotone operators and the operators of $(M)$ type have the following inclusion relation.

Lemma 2.3. (^[18]) Let $A$ be an operator mapping $V$ to $V'$; if $A$ is a bounded semi-continuous monotone operator, then $A$ is an operator of $(M)$ type.

The next lemma is a corollary of Schauder's fixed point theorem to prove the existence of stationary solutions to the problem (1.6).

Lemma 2.4. (^[36]) Let $X$ be a finite-dimensional Hilbert space equipped with scalar product $[\cdot, \cdot]$ and norm $[\cdot]$. $F:X\rightarrow X$ is a continuous mapping. If there exists $k > 0$, such that

then there exist $\xi \in X$ and $[\xi] \leq k$, such that $F(\xi) = 0$.

2.2.   Statement of the problem

In virtue of the above operators $P$, $A$, and $A_1$, we can transform the problem (1.6) into the following equivalent abstract form

We will establish the well-posedness and stability results for the problem (2.7) with infinite delay operators in the following phase:

which is a Banach space endowed with the norm

In order to obtain the existence and uniqueness of solutions to the problem (2.7), we impose some appropriate assumptions on delay term $g$.

(H-g) Assume that $g:[0, T]\times BCL_{-\infty}(H)\rightarrow (L^2(\Omega))^3$ satisfies the following:

$(g1)$ $g(t, 0) = 0, \quad \forall t \in [0, T]$.

$(g2)$ For all $\xi\in BCL_{-\infty}(H)$, the mapping $[0, T]\ni t\rightarrow g(t, \xi) \in (L^2(\Omega))^3$ is measurable.

$(g3)$ There exists $L_g > 0$ such that for all $\xi, \eta \in BCL_{-\infty}(H)$,

Remark 1. (i) As pointed out in ^[28,32], $(g1)$ is not a restriction. Indeed, if $g(t, 0)\neq 0$, we could redefine $\bar{f}(t) = f (t) + g(t, 0)$ and $\bar{g}(t, \cdot) = g(t, \cdot)-g(t, 0)$. At this time, $\bar{g}$ meets the condition $(g1)$, and the problem is not changed in this way.

(ii) The conditions $(g1)$ and $(g3)$ indicate that for any $\xi \in BCL_{-\infty}(H)$,

Thus, it holds that $|g(\cdot, \xi)|_2 \in L^{\infty}(0, T)$.

(iii) In particular, examples of a distributed delay operator and a variable delay operator satisfying conditions $(g1)$–$(g3)$ can be given respectively (refer to Examples 2.2 and 2.4 in ^[32]).

Above all, we establish the following definition of global weak solutions to the problem (2.7).

Definition 2.5. Let $T > 0$ and the initial datum $\phi\in BCL_{-\infty}(H)$. A weak solution to the problem (2.7) in $(-\infty, T]$ is a function $u = u(t, x) \in C((-\infty, T];H) \cap L^4 (0, T;V)$ such that $u_0 = \phi$ and

holds for all $v \in V$ in the sense of $D'(0, T)$. $D'(0, T)$ denotes the distribution space composed of functions, and it is defined on $(0, T)$ and valued in ${\mathbb{R}}$.

Remark 2. If $u$ is a weak solution to the problem (2.7), then $u$ satisfies the following energy equality

3.   Well-posedness and regularity of global solutions

In this section, we prove the existence, uniqueness and regularity of weak solutions to the problem (2.7). The following theorem is one of the main results.

Theorem 3.1. ${\bf1)}$ Consider that $g$ satisfies the assumptions encompassed by (H-g). If $f \in L^{\frac{4}{3}}(0, T;V')$ and $\phi \in BCL_{-\infty}(H)$, then the problem (2.7) possesses a unique weak solution $u \in C((-\infty, T];H)\cap L^4(0, T;V)$.

2) Moreover, if $f \in L^{2}(0, T;(L^2(\Omega))^3)$ and $\phi \in BCL_{-\infty}(H)$ with $\phi(0) \in V$, the weak solution $u$ is strong. That is, $u \in C([0, T];V)\cap L^2(0, T;D(A))$.

Proof. We split the proof into several steps.

Step 1: Galerkin scheme and a priori estimates

By applying the classical spectral theory of the elliptic operators, let $\{\omega_i\}^\infty_{i = 1} \in V$ be the basis of all eigenfunctions for the Stokes operator $A$, which is also a complete orthonormal basis in $H$ and $V$. Denote $V_m = \text {span}\{\omega_1, \cdot\cdot\cdot, \omega_m\}$ and consider the projector $P_m:H\rightarrow V_m$ given by

We define the approximated solution $u_m(t) = \sum\limits_{i = 1}^{m}h_{im}(t)\omega_i$ that satisfies following Cauchy problem:

The problem (3.1) is equivalent to a set of functional differential equations with infinite delay and the unknown variable $\{h_{1m}(t), h_{2m}(t), \cdot \cdot \cdot, h_{mm}(t)\}$. From Theorem 1.1 in ^[37], it can be obtained that the problem (3.1) has a unique local solution $u_m$ on the interval $[0, t_m].$

Next, we will obtain that $u_m$ exists globally by proving a prior estimate. Multiplying $(3.1)_1$ by $h_{im}(t)$, and then summing $i$ from $1$ to $m$, with the help of the Cauchy-Schwartz inequality and Young's inequality, we have

For convenience, we use $\mu$ to denote $(\frac{27}{16\nu_1})^{\frac13}$. Integrating (3.2) with respect to the time variable $t$ from $0$ to $t$, we derive

which implies that

Applying the Gronwall inequality to (3.4), we get

i.e.

where $C(R, T)$ is a constant that is dependent on $\nu_1, L_g, T, R > 0$ and $f$.

Further, from (3.3) and (3.5), it follows that

Considering that $f \in L^{\frac{4}{3}}(0, T;V')$, there exists another constant $C(R, T)$ (relabeled to be the same) such that

Hence, (3.5) and (3.6) indicate that

Observe that $(3.1)_1$ is equivalent to the following:

Combining Remark 1(ii) and (3.5), one has

Notice that $\|P_m\|_{\mathscr{L}(V, V)}\leq1$ and $P_m^* = P_m$, so $\|P_m\|_{\mathscr{L}(V', V')}\leq 1$. Thus, it follows from (2.4), $(2.5)_1$ and (3.7)–(3.9) that

Step 2: Compactness results and approximations in $BCL_{-\infty}(H)$ for the initial datum

By (3.1), (3.6)–(3.10) and the Aubin-Lions compactness lemma, we deduce that there exists a subsequence (still denoted by $\{u_m\}$) and $u \in L^\infty(0, T;H) \cap L^4 (0, T;V)$ such that, when $m\rightarrow +\infty$,

Thanks to $(3.11)_3$ and $(3.11)_4$, we infer that $u_m, u \in C([0, T];H)$. Next, we verify that the initial value sequence satisfies

In consideration of $\phi \in BCL_{-\infty}(H)$, we set $\lim\limits_{\theta\rightarrow -\infty}\phi(\theta) = \tilde{\phi} \in H$. That is to say, for any $\varepsilon > 0$, there is a large enough $N > 0$ such that

Since $P_m$ is a projection operator, we can find an $M_1 > N$ such that

Observe that, for any $\theta \in [-N, 0]$, $|P_m\phi(\theta)- \phi(\theta)|_2$ is non-increasing with respect to $m$, and, for almost everywhere where $-N \leq \theta \leq 0$,

Also, it holds that

Based on one Dini's theorem (i.e. Lemma 2.1), there exists an $M_2 > 0$ such that

Now, we conclude from (3.14) and (3.15) that

Thus, we complete the proof of (3.12).

Step 3: Energy method and the existence of solutions

To take the limit of the approximate system (3.1), we will prove that the nonlinear term can exceed the limit. For the nonlinear convection term $b(u_m, u_m, \omega_i)$, the same procedure as in ^[38] can yield

Regarding the nonlinear viscosity term $\langle A_1u_m, \omega_i\rangle$, we use a similar technique as in ^[39]. Notice that here we can only obtain the weak convergence of the infinite delay term $g(t, u_{mt})$, that is,

However, by using the Hölder inequality, we have

which, along with $(3.11)_1$ and (3.17), yields

Then, following the same proof process as equation (28) in our previous work ^[38], we have

Because the delay $g(t, u_{mt})$ is infinite here, we also need to prove that the limit can exceed it. Next, we will use the energy method to obtain following limit process:

On account of the condition $(g3)$, we have

Thus, in order to prove (3.20), we only need to prove that, for all $0 \leq t \leq T$,

From the definition of the space $BCL_{-\infty}(H)$, we can derive

Noting the convergence of (3.12), we only need to discuss the second term on the right-hand side of (3.23), i.e.,

First of all, taking into account $(3.11)_1$, it follows that

However, this is not enough to prove (3.24). By applying the Hölder inequality, we can infer the following:

Hence, (3.26) and (3.10) show that $\{u_m(t)\}$ is uniformly equicontinuous on $[0, T]$ in $V'$. Given the fact that $H\hookrightarrow V'$ is compact and (3.5), we can see that for any $t \in[0, T], \{u_m(t)\}$ is a precompact set in $V'$. Then by the Arzelà-Ascoli theorem, one has

Again from (3.7), we obtain that for any sequence $\{t_m\} \in[0, T]$ with $\lim\limits_{m\rightarrow \infty}t_m = t$,

where we have used (3.27) to identify which is the weak limit.

Next, we are going to prove (3.24) by a contradiction argument. If (3.24) does not hold, considering $u\in C([0, T]; H)$, then we have that $\varepsilon > 0$ and $t_0\in [0, T]$, as well as the subsequences $\{u_m(t)\}$ (still relabelled with the same notation) and $\{t_m\} \subset [0, T]$ with $\lim\limits_{m\rightarrow \infty}t_m = t_0$, such that

We will prove that this is incorrect by using the energy method. From (3.2), it can be seen that for all $u_m$, there is the following energy inequality:

where $C'' = \frac{D}{2\nu_0\lambda_1}$ and $D$ corresponds to the upper bound given by

Given (3.11), (3.16) and (3.19), taking the limit of (3.1), we deduce that

and $u(0) = \phi(0)$. If $v = u$ in (3.31), the following energy equality holds

On the other hand, by $(3.11)_6$ and the weak lower semicontinuity of norms, it can be concluded that

Then, by using (3.32), (3.33), combined with the Cauchy-Schwartz inequality and (2.2), it follows that for any $0\leq s \leq t\leq T$,

So, we see that $u$ also satisfies the energy inequality (3.30) with the same constant $C''$.

Now, we consider the functions $J, J_m: [0, T] \rightarrow \mathbb{R}$ respectively defined by

On account of (3.30) and (3.34), it is clear that $J$ and $J_m$ are continuous and non-increasing functions on $[0, T]$. We infer from $(3.11)_3$ and (3.25) that $J_m(t)\rightarrow J(t) \text{ alomst everywhere that } t\in [0, T].$

Thanks to (3.28), we have that

Therefore, it holds that

We obtain from (3.35) and (3.36) that

Thanks to (3.35) and (3.37), one has

which contradicts (3.29); so, we get (3.24). Further, (3.22) and (3.20) are also true. Now, based on the limit of (3.1), by combining (3.16), (3.19) and (3.20), we can infer that $u$ is indeed a weak solution to the problem (2.7).

Finally, it is necessary to prove that (3.36) is valid. We discuss $t_0$ in the following two cases:

1) If $t_0 = 0$, let $s = 0$ and $t = t_m$ in (3.30); then, taking the upper limit of both ends, by $\lim\limits_{m\rightarrow \infty}t_m = 0$ and $u_m(0) = P_m\phi(0) = P_mu(0)$, we obtain

namely, (3.36) is true. Only when $0 < t_0\leq T$ can we take the approximation sequence from the left so that its limit is $t_0$; so, a separate discussion of $t_0 = 0$ is necessary here.

2) If $0 < t_0\leq T$, we can take the sequence $\{t'_k\} \subset (0, t_0)$ such that $\lim\limits_{k\rightarrow \infty}t'_k = t_0$ and $\lim\limits_{m\rightarrow \infty}J_m(t'_k) = J(t'_k)$ for all $k$. From the continuity of $J(s)$, it can be deduced that for any $\varepsilon > 0$, there exists $k_\varepsilon \in \mathbb{N}$ such that

Because $\lim\limits_{m\rightarrow \infty}t_m = t_0$ and the sequence $\{t'_k\}$ tends to $t_0$ from the left, we can take $m(k_\varepsilon)$ such that, for all $m\geq m(k_\varepsilon)$,

According to the non-increasing property of $J_m$, for any $m \geq m(k_\varepsilon)$, we can obtain from (3.38) and (3.39) that

By the arbitrariness of $\varepsilon$, (3.40) yields that $\limsup\limits_{m\rightarrow \infty}J_m(t_m)\leq J(t_0)$, i.e.,

Thanks to $(3.11)_3$, we know that

Then, by using (3.41) and (3.42), we have

So (3.36) is proved. From the above two cases, we deduce that (3.36) is valid for any $0\leq t_ 0 \leq t$. Thus, according to the above analysis, we have proved the existence of weak solutions.

Step 4: Uniqueness of solution

If $u$ and $v$ are two solutions of problem (2.7) with the same initial value $\phi$, and if $\omega(t) = u(t)-v(t)$, then we have

Note that

Taking the inner product of (3.43) and $\omega(t)$, we can derive from (3.44) and $(2.5)_2$ that

Because of conditions $(g_3)$ and the monotonicity of $A_1$, with the help of $(2.5)_4$ and the Young inequality, it follows that

Denote $\mu_1 = \frac{27c_1^4}{32\nu_0^3}$. From $\omega(0) = 0$ and the definition of the norm of $BCL_{-\infty}(H)$, one has

Since $\omega(\theta) = 0$ for any $\theta \leq 0$, (3.46) indicates that

which, together with the Gronwall inequality, yields that $\omega\equiv 0$, i.e., the solution is unique.

Step 5: Regularity of the weak solution

At this time, we assume that the non-delay external force $f \in L^{2}(0, T;(L^2(\Omega))^3)$ and the initial value satisfies that $\phi \in BCL_{-\infty} (H)$ with $\phi(0) \in V$. Multiplying $(3.1)_1$ by $\lambda_i h_{im}(t)$, and then summing from $i = 1$ to $i = m$, in view of $A\omega_i = \lambda_i\omega_i$ and Young's inequality, we have

In particular, it can be derived from $(2.5)_4$ and Young's inequality that

where $\mu_2 = \frac{27c_1^8}{32\nu_0^3\nu_1^4}$. Integrating (3.47) from $0$ to $t$, we obtain from (3.48) that

By $\phi(0) \in V$, $f \in L^{2}(0, T;(L^2(\Omega))^3)$, (3.9) and the fact that $\|u_m (0)\| = \|P_m \phi(0)\| \leq \|\phi(0)\|$, we conclude that

Moreover, applying Agmon's inequality ($\|u\|_{L^\infty(\Omega)} \leq c_2\|u\|^{\frac12}|Au|_2^{\frac12}, \ \forall u \in D(A)$) to the convection term, one has

where the last inequality uses (3.50). Thus, $B(u_m, u_m) \in L^2(0, T;H)$. For the nonlinear viscosity, we can get

Then, using a similar process as in Step 1, we deduce that

Combining (3.50), (3.53) and the Aubin-Lions compactness lemma, it follows that $u \in C([0, T];V)\cap L^2(0, T;D(A))$. Therefore, the weak solution obtained above is indeed strong. □

4.   Stability of stationary solutions

In this section, we shall consider the existence and stability of stationary solutions to the problem (2.7) under some appropriate assumptions.

4.1.   Existence and uniqueness of stationary solutions

In this subsection, by using a corollary of Schauder's fixed point theorem, we establish the existence and uniqueness of the stationary solution to the problem (2.7).

In order to investigate the existence and properties of stationary solutions to (2.7), we assume that $f$ is independent of time, i.e., $f \in V'$. Assuming that $\rho \in C^1(\mathbb{R}_+, \mathbb{R}_+)$ and $\sup\limits_{t\geq 0}\rho'(t) = \rho_* < 1$, $g(t, u_t) = G(u(t-\rho(t))): H\rightarrow (L^2(\Omega))^3$ satisfies the following conditions (see ^[32,33]):

$(G1)$ $G(0) = 0$;

$(G2)$ There exists $L_G > 0$, such that for any $\xi, \eta \in H$,

Under the above assumptions, the stationary equation of the problem (2.7) has the following form:

A function $u^* \in V$ is called a weak solution to (4.1) if it satisfies that

And, $u^*$ is also called the stationary solution of the problem (2.7).

Theorem 4.1. Assume that $f \in V'$, $g(t, u_t) = G(u(t-\rho(t)))$ satisfies conditions $(G_1)$ and $(G_2)$ and $\nu_0 > \frac{L_G}{\lambda_1}$; then, there exists at least one weak solution $u^*$ to (4.1) satisfying

In addition, if $(\nu_0-\lambda_1^{-1}L_G)^2 > c_1\lambda_1^{-\frac14}\|f\|_*$ holds, the solution is unique.

Proof. First, we prove the existence of weak solutions. Let $\{v_j\}^\infty_{j = 1}$ be the orthonormal basis of $V$, composed of the characteristic function of Stokes operator $A$. Set $V_m = \text {span}\{v_1, \cdot\cdot\cdot, v_m\}$; then we respectively construct the approximate solution and the approximate system as follows:

Since system (4.5) is nonlinear, the existence of $\{h_{1m}, \cdot\cdot\cdot\, h_{mm}\}$ is not obvious, that is, we need to prove the existence of the approximate solution $u_m$. We define $R_m:V_m\rightarrow V_m$ as follows: for all $u, v \in V_m$,

Obviously, $R_m$ is continuous. Furthermore, we have the following for any $u \in V_m$ and $u \neq 0$:

By taking $\|u\| = \beta = \frac{\|f\| _*}{\nu_0-L_G\lambda_1^{-1}}$, we obtain that $((R_mu, u))\geq0$. Thus, according to a corollary of Schauder's fixed point theorem (namely, Lemma 2.4), for each $m\geq 1$, there exists $u_m \in V_m$ such that $R_m(u_m) = 0$ and $\|u_m \| \leq \beta$, i.e.,

So sequence $\{u_m\}$ is uniformly bounded in $V$. Because the embedding $V\hookrightarrow H$ is compact, there exists a subsequence (still recorded as $\{u_m\}$), which converges weakly in $V$ and strongly converges to an element $u^* \in V$ in $H$.

In order to exceed the limit of (4.5), it is necessary to discuss the nonlinear term. Using similar methods in Theorem 3.1, it can be proved that the convection term can exceed the limit. For the nonlinear term $A_1u_m$, because $A_1$ is a bounded semi-continuous monotone operator and Lemma 2.3 holds, $A_1$ is an operator of $(M)$ type. In addition, the above results show that

By taking the limit of Eq (4.7), $\chi = f+G(u^*)- \nu_0Au^* -B(u^*, u^*)$. We multiply (4.5) by $h_{jm}(t)$, and sum $j$ to obtain

By virtue of (4.8), one has

Using the weak lower semi-continuity of norms, one has

It follows from the condition $(G2)$ that

which, together with

gives

Combining (4.10)–(4.12) with (4.14), we get

Since $A_1$ is an operator of type $(M)$, we infer from (4.8), (4.9) and (4.15) that

Now, taking the limit of (4.5) and combining the convergences given by (4.8), (4.13) and (4.16), we know that $u^*$ is indeed a weak solution to problem (4.1), and it satisfies that $\|u^*\|\leq\frac{\|f\| _*}{\nu_0-L_G\lambda_1^{-1}}.$ Next, we prove the uniqueness of the solutions. Let $u^*$ and $v^*$ be the two solutions to (4.1). Set $w = u^*-v^*$; then,

Taking the inner product of (4.17) by $w$, by the monotonicity of $A_1$ and $(G2)$, we deduce that

Because of $u^* \leq \beta$,

which, together with $(\nu_0-\lambda_1^{-1}L_G)^2 > c_1\lambda_1^{-\frac14}\|f\|_*$, implies that $w = 0$, that is, the solution is unique. Thus, we have completed the proof of the theorem. □

4.2.   Local stability of stationary solutions

In this subsection, we prove the local stability of the stationary solution obtained in Theorem 4.1. In order to analyze the stability of stationary solutions to problem (2.7), we first review the definition of stability as follows (see ^[32,33] for details).

Definition 4.2. A stationary solution $u_\infty$ to (2.7) is stable if for any $\varepsilon > 0$ there exists $\delta > 0$ such that, if $\varphi \in BCL_{-\infty} (H)$ and $\|\varphi-u_\infty\|_{BCL_{-\infty}(H)} < \delta$, then the solution $u(\cdot, \varphi)$ to (2.7) exists for all $t \geq 0$ and satisfies

A stationary solution $ u_\infty $ to (2.7) is said to be asymptotically stable if it is stable and the solution $u(\cdot, \varphi)$ to (2.7) exists for all $t \geq 0$ and satisfies

Theorem 4.3. Consider that $f \in V'$, $g(t, u_t) = G(u(t-\rho(t)))$ satisfies the assumptions $(G_1)$ and $(G_2)$, and $\nu_0$ satisfies that $\nu_0 > \frac{L_G}{\lambda_1}$; then, (4.1) has a weak solution $u_\infty$ satisfying (4.3). Moreover, if

then the solution $u_\infty$ is unique and, for any $\phi \in BCL_{-\infty}(H)$, the solution $u(t)$ to (2.7) with $f(t)\equiv f$ satisfies

where $C_1 = \max \{1, (\frac{L_G}{1-\rho^*})^\frac12\}$. Namely, the stationary solution $u_\infty$ to the problem (2.7) is stable.

Proof. First, it is easy to verify that $(\nu_0-\lambda_1^{-1}L_G)^2 > c_1\lambda_1^{-\frac14}\|f\|_*$ by using (4.20). Thus, from Theorem 4.1, we can know that problem (4.1) has a unique weak solution $u_\infty$ satisfying

Using Theorem 3.1, under the above assumptions, the problem (2.7) has a unique weak solution, which is recorded as $u(t)$. Set $w = u(t)-u_\infty$; then, it is immediate that

Note the following fact

Taking the inner product of (4.22) and $w$, it is easy to obtain the following from the Young inequality and (4.21):

Integrating (4.23) with respect to the time, with the help of a change of variable in the integral of $\rho$ and (4.20), we deduce that

Therefore, by taking $C_1 = \max \{1, (\frac{L_G}{1-\rho^*})^\frac12\}$, one has

Thus, the proof is completed. □

4.3.   Asymptotic stability of stationary solutions

Theorem 4.3 only gives the local stability of stationary solutions to the problem (2.7), and we have not demonstrated the asymptotic stability of stationary solutions. In this subsection, the asymptotic stability of stationary solutions will be proved under two specific conditions for delayed forces.

1). Under the assumption of Theorem 4.3, we additionally require that the delay interval of external force $G$ is bounded. That is, $\rho(t) \in [0, h]$, where $h > 0$ is a constant. At this time, using the Lyapunov function method, we obtain that the stationary solution is exponentially stable.

Theorem 4.4. Under the assumption of Theorem 4.3, if $\rho(t)$ satisfies that $\rho(t) \in [0, h]$, then there exists a sufficiently small positive constant $\lambda$ such that for any $t\geq 0$,

where $C_2 = \max \{1, (\frac{L_Ge^{\lambda h}}{1-\rho^*})^\frac12\}$.

Proof. By a process similar to the proof of Theorem 4.3, multiplying both sides of (4.23) by $e^{\lambda t}$, we obtain

Integrating (4.25) in time, we get

In order to estimate the delay term, by transforming $\tau = s -\rho(s) = \theta(s)$, it is easy to deduce that

Because $\theta$ is a monotonic increasing function and $\rho (t) \in [0, h]$, $\theta^{-1}(\tau) \leq \tau +h$ for any $\tau \geq -\rho (0)$; then,

By combining (4.26) and (4.27), we conclude that

In view of the assumption (4.20), there exists a sufficiently small $\lambda > 0$ such that

Hence, we obtain

where $C_2 = \max \{1, (\frac{L_Ge^{\lambda h}}{1-\rho^*})^\frac12\}$. Hence, we complete the proof of Theorem 4.4. □

2). Under the assumption of Theorem 4.3, we additionally require that external force $G$ be a special case of proportional delay. That is, $\rho(t) = (1-q)t$ for $0 < q < 1$. At this point, we will give a sufficient condition for the trivial stationary solution to have polynomial stability. Here, we use a lemma of the proportional equation

Lemma 4.5. (^[40], Lemma 3.6) Let $a < 0, b > 0$ and $q \in (0, 1)$. If $h \in C(\mathbb{R}_+, \mathbb{R}_+)$ satisfies that for any $t \geq 0$, $D^+ h(t) \leq ah(t) + bh(qt)$ and $h(0) > 0$, here,

then, there exists a constant $C = C(a, b, q) > 0$ such that

where $\gamma$ satisfies that $a + bq^\gamma = 0$.}

In particular, using the lemma above, we can obtain a polynomial stability result for the trivial stationary solution to the problem (2.7).

Theorem 4.6. Consider (2.7) with $f \equiv 0, g(t, u_t) : = L_Gu(qt)$ with $0 < q < 1$, $L_G \in \mathbb{R}$ and $\nu_0 > \frac {|L_G|}{\lambda_1}$. Then the origin is the only stationary solution to the problem. Moreover, for any solution to (2.7) given the initial data $\phi \in BCL_{-\infty}(H)$ with $\phi(0)\neq 0$, there exists a positive constant $C_3$ that id dependent on $L_G, \nu_0, \lambda_1$ and $q$ such that

where

Proof. The existence and uniqueness of $u_\infty$ can be obtained from Theorem 4.1. In addition, it is easy to know that the origin satisfies the conditions of the stationary equation, so $u_\infty\equiv 0$. Taking the inner product of (2.7) and $u$ in the space $H$, we get

By virtue of the Poincaré inequality and Young's inequality, it is immediate that

In order to use the Lemma 4.5, set $h(t) = |u(t)|_2^2$. It holds from (4.33) that

and $h(0) = |u(0)|_2^2 = |\phi(0)|_2^2 > 0$. From $\nu_0 > \frac {|L_G|}{\lambda_1}$, it is evident that $|L_G|-2\lambda_1\nu_0 < 0$ and $|L_G| > 0$. Therefore, according to Lemma 4.5, there exists a constant $C_3(L_G, \nu_0, \lambda_1, q) > 0$ such that

i.e.,

where $\gamma$ satisfies that $|L_G|-2\lambda_1\nu_0 + |L_G|q^\gamma = 0$. That is, $\gamma$ is given by $\gamma = \log_q(\frac {2\lambda_1\nu_0-|L_G|}{|L_G|})$. Since $q\in (0, 1)$, one has that $\gamma < 0$. This completes the proof. □

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are grateful to the referees for their helpful suggestions which improved the presentation of this paper. This research was supported by the NSFC (No. 11501199).

Conflict of interest

The authors declare that there is no conflicts of interest.